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This paper presents the lowest-order weak Galerkin finite element method for linear elasticity problems on the convex polygonal meshes. This method uses piecewise constant vector-valued spaces on element interiors and edges. The discrete weak gradient space introduced by this paper is the matrix version of C W 0 space. The discrete weak divergence space is piecewise constant space on each element. This method is simple, efficient, stabilizer-free and symmetric positive-definite. The optimal error estimates in discrete H 1 and L 2 norms are presented. Numerical results are given to demonstrate the efficiency of algorithm and the locking-free property. • The matrix version of C W 0 element for discrete weak gradient is introduced. • The lowest-order weak Galerkin finite element space ( P 0 2 , P 0 2 , C W 0 2 , P 0 ) is adopted. • Our method is suitable for the polygonal and hybrid meshes. [ABSTRACT FROM AUTHOR]

In this paper, a novel and fast two-grid finite element method (FEM) is proposed for efficiently solving semilinear time-fractional diffusion equations with variable coefficients. To handle the initial singularity, the nonuniform L1 scheme is employed in the temporal domain. A novel two-grid FEM technique is used in the spatial domain to reduce computational cost. The two-grid algorithm solves the original nonlinear fractional equation on a much coarser grid with size H , obtaining the coarse-grid solution u H n based on the fine-grid solutions at previous time levels. This approach avoids repetitive discrete convolutional summation when approximating the Caputo derivative on the coarse grid, resulting in reduced computational cost. Furthermore, a fast nonuniform L1 scheme with two-grid technique is developed to accelerate the evaluation of the Caputo derivative. The paper also proposes a novel, fast and highly accurate algorithm for computing the coefficients involved in the discretized Caputo fractional derivative. The α -robust stability and optimal L 2 - and H 1 -norm error estimates for fully discrete scheme are derived, where the error bound remains valid as α → 1 − . Numerical results validate the theoretical findings and demonstrate the superior efficiency of the proposed two-grid algorithms compared to the standard FEM. • A novel, fast two-grid FEM is proposed for time-fractional diffusion equations. • A innovative method for computing the discrete convolution kernels is proposed. • The α -robust stability and optimal L 2 - and H 1 -norm error estimates are derived. • Numerical results show the efficiency and accuracy of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the convergence of an adaptive two-grid weak Galerkin (ATGWG) finite element method for second order semilinear elliptic partial differential equations (PDEs). First, we propose an ATGWG method and then prove that the sum of the energy error and the error estimator of ATGWG method between two consecutive adaptive loops is a contraction. The weak Galerkin (WG) elements (P j (T) , P ℓ (∂ T) , R T j (T)) (Wang and Ye, 2013) are studied in this paper and numerical experiments based on the lowest order case with j = l = 0 are provided to support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the unconditional stability and optimal error estimate of the velocity, pressure and angular velocity for the modified characteristics FEMs of the unsteady Micropolar Naiver–Stokes Equations (MNSE) are presented. In this method, the nonlinear equation is linearized by the characteristic finite element method for dealing with the time derivative term and the convection term. Basing on the characteristic time-discrete system, the error function is split into a temporal error and a spatial error. With a rigorous analysis to the characteristic time-discrete system, we prove that the error between the numerical solution and the solution of the time-discrete system is τ -independent, where τ denotes the time stepsize. The stability results and optimal error estimates in L 2 norm and H 1 norm will be given. Finally, some numerical results will be provided to confirm our theoretical analysis. • The unconditional stability and optimal error estimate for the modified characteristics FEMs are presented. • The nonlinear equation is linearized by the characteristic method. • Basing on the characteristic time-discrete system, the error function is split into a temporal error and a spatial error. • The stability results and optimal error estimates in $L̂2$ norm and $Ĥ1$ norm will be given. [ABSTRACT FROM AUTHOR]

In this paper, based on the scalar auxiliary variable (SAV) approach, the superconvergence error analysis is investigated for the time-dependent Navier–Stokes equations. In which, an equivalent system of the Navier–Stokes equations with three variables and a fully-discrete scheme is developed with semi-implicit Euler discretization for the temporal direction and low-order bilinear-constant finite element discretization for the spatial direction, respectively. With the help of the high-precision estimations of the bilinear-constant finite element pair on the rectangular meshes, the superclose error estimates for velocity in H 1 -norm and pressure in L 2 -norm are obtained by treating the trilinear term carefully and skillfully. The global superconvergence results are also derived in terms of a simple and efficient interpolation post-processing technique. Finally, some numerical results are provided to demonstrate the correctness of the theoretical analysis. • A semi-implicit SAV Galerkin FEM is investigated for the time-dependent N–S equations. • The high-precision estimations are employed in the error analysis. • The trilinear term is treated skillfully in the error analysis. • The superclose and superconvergence results are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we propose a second-order decoupled finite element method based on lagging a part of the interfacial coupling terms for the time dependent Stokes–Darcy problem, which only need to solve two sub-physical problems sequentially. Under a modest time step restriction Δ t ≤ C (physical parameters), the optimal long time error estimates are obtained both in the L 2 norm and in the H 1 norm. Numerical results are provided to illustrate the convergence rate O (Δ t 2) of the temporal approximation. • A second-order decoupled finite element method based on lagging a part of the interfacial coupling terms is proposed. • The optimal long time error estimates are obtained in the L 2 norm. • The optimal long time error estimates are obtained in the H 1 norm. [ABSTRACT FROM AUTHOR]

This paper presents a Crank–Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. The spatial discretization adopts the Galerkin finite element method (FEM), and the temporal discretization employs the CNLF method for linear terms and the semi-implicit method for nonlinear terms. The first step uses Stokes style's scheme, the second step employs the Crank–Nicolson extrapolation scheme, and others apply the CNLF scheme. We establish that the fully discrete scheme is stable and convergent when the time step is less than or equal to a positive constant. Firstly, we show the stability of the scheme by means of the mathematical induction method. Next, we focus on analyzing error estimates of the CNLF method, where the convergence order of the velocity and magnetic field reach second-order accuracy, and the pressure is the first-order convergence accuracy. Finally, the numerical examples demonstrate the optimal error estimates of the proposed algorithm. • Crank–Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. • The CNLF method for linear terms and the semi-implicit method for nonlinear terms. • Fully discrete scheme is stable and convergent. • The convergence order of the velocity and magnetic field reach second-order accuracy. [ABSTRACT FROM AUTHOR]

This paper considers the finite element method to solve a generalized constant delay diffusion equation. The regularity of the solution of the considered model is investigated, which is the first time to discover that the solution has non-uniform multi-singularity in time compared with Tan et al. (2022). To overcome the multi-singularity, a symmetrical graded mesh is used to devise the fully discrete finite element scheme for the considered problem based on L1 formula of the Caputo fractional derivative and fractional trapezoidal formula of the Riemann–Liouville fractional integral. Then we investigate the unconditional stability of this scheme. Next, the local truncation errors of the L1 formula and the fractional trapezoidal formula are analyzed in detail, especially the later one is discussed at the first time, under the multi-singularity of the solution and the symmetrical graded mesh. Using these error results, we obtain the convergence of the proposed numerical scheme. Finally, some numerical tests are provided to verify the obtained theoretical results. • The regularity of the solution is investigated and shows multi-singularity in time. • A symmetrical graded mesh is used to overcome the multi-singularity. • The unconditional stability and optimal temporal convergence rate are obtained. • The truncation errors of L1 formula and fractional trapezoidal formula are discussed. • Some numerical tests are provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents dual-robust iterative algorithms for the 2D/3D steady thermally coupled inductionless magnetohydrodynamics (IMHD) system in a general Lipschitz domain. Both velocity and current density are discretized by the divergence-conforming elements. Furthermore, we utilize an interior penalty discontinuous Galerkin (IPDG) approach to guarantee the H 1 -continuity of velocity. With the system's strong nonlinearity, we propose three iterative algorithms (Stokes, Newton and Oseen iterations) and provide analytical proofs for their stability and convergence. And the feature of these methods is that they simultaneously ensure the complete divergence-free of discrete velocity and discrete current density. Finally, the numerical simulations verify theoretical analysis and the effectiveness of proposed algorithms. • H(div)-conforming IPDG finite element method. • The approximations for velocity and current density are exactly divergence-free. • Iterative analysis based on pressure robustness and electric potential robustness. • The theoretical analysis is validated by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose two Multipoint Stress Mixed Finite Element (MSMFE) methods for linear viscoelasticity with weak symmetry on quadrilateral grids. The methods are constructed based on the lowest order Brezzi–Douglas–Marini mixed finite element spaces for elastic and viscous stress, piecewise constant velocity and piecewise constant (linear) vorticity. A special quadrature rule is applied for local stress and vorticity elimination. This results in a positive definite cell centered velocity–vorticity or only velocity system at each time step. Unconditional energy-dissipation of the MSMFE methods is proved rigorously. The accuracy of all the numerical solutions in their nature norms are established to be first order space convergence, both for the semi-discrete and fully-discrete formulations. Numerical results validate the theory results of the proposed methods. • We propose two efficient multipoint stress mixed finite element methods. • The resulting system can be deduced as positive definite cell-centered velocity–vorticity or only velocity system. • The accuracy of all the numerical solutions in their nature norms are established. • Several numerical examples validate the theory results. [ABSTRACT FROM AUTHOR]

This paper develops a weak Galerkin (WG) finite element method of arbitrary order for the steady incompressible Magnetohydrodynamics equations. The WG scheme uses piecewise polynomials of degrees k (k ≥ 1) , k , k − 1 and k − 1 respectively for the approximations of the velocity, the magnetic field, the pressure, and the magnetic pseudo-pressure in the interior of elements, and uses piecewise polynomials of degree k for their numerical traces on the interfaces of elements. The method is shown to yield globally divergence-free approximations of the velocity and magnetic fields. We give existence and uniqueness results for the discrete scheme and derive optimal a priori error estimates. We also present a convergent linearized iterative algorithm. Numerical experiments are provided to verify the obtained theoretical results. • We develop a weak Galerkin method of arbitrary order for the steady incompressible MHD equations. • The method is shown to yield globally divergence-free approximations of the velocity and magnetic fields. • We give existence and uniqueness results for the discrete scheme and derive optimal a priori error estimates. • We also present a convergent linearized iterative algorithm. • Numerical experiments confirm the theoretical results. [ABSTRACT FROM AUTHOR]

This paper aims to study an H 1 -Galerkin mixed finite element method (FEM) for Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity. The bilinear element and zero-order Raviart–Thomas element ( Q 11 / Q 10 × Q 01 ) are taken as approximation spaces for the original variables p , ϕ and the auxiliary variables u = ∇ p and ψ = ∇ ϕ , respectively. The supercloseness and superconvergence estimates of order O (h 2 + τ 2) are presented for p and ϕ in the H 1 norm and u and ψ in the L 2 norm, which improve the known results that the original variable ϕ can only reach optimal error estimate shown in numerical experiments without theoretical analysis. Here h is the subdivision parameter and τ is the time step. Finally, some numerical results are provided to support the theoretical analysis. • An H 1 -Galerkin mixed FEM for nonlinear KGZ equations is studied. • The superconvergence estimates are presented, which improve the known results. • The results are also valid to the nonconforming FE pair E Q 1 r o t / Q 10 × Q 01 . [ABSTRACT FROM AUTHOR]

In this paper, we investigate a fully discrete finite element scheme for the incompressible magnetohydrodynamic (MHD) equations with magnetic-current formulation that was introduced in Hu et al. (2016). We discretize the system by the semi-implicit Euler scheme in time and a mixed finite element approach together with finite element exterior calculus in space. The resulting scheme enjoys the structure-preserving feature that it can always produce an exactly divergence-free magnetic induction on the discrete level. The unique solvability and unconditional stability of the scheme are also proved rigorously. By utilizing the energy argument, error estimates for the velocity, magnetic induction, current density and induced electric field are further established under the low regularity hypothesis for the exact solutions. Numerical results are provided to verify the theoretical analysis and to show the effectiveness of the proposed scheme. • Error analysis of a fully discrete finite element method for the MHD equations is given. • The proposed scheme is linear, unconditional stability and structure-preserving. • Numerical examples are provided to verify the theoretical findings. [ABSTRACT FROM AUTHOR]